Math

Quadratic Equation Calculator

What this does

Solve ax² + bx + c = 0 for real or complex roots, showing the discriminant and vertex so the answer never arrives unexplained.

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Calculator inputs

How to use it

Using the quadratic equation calculator

  1. 01

    Enter all three coefficients

    Include signs: x² − 5x + 6 means a=1, b=−5, c=6.

  2. 02

    Check the discriminant row

    It predicts two roots, one double root, or a complex pair before you read them.

  3. 03

    Read the roots

    Real solutions appear as plain numbers; complex ones as real ± imaginary·i.

Good to know

The discriminant is the story

Before any root appears, b² − 4ac already tells you whether the parabola slices the x-axis twice, kisses it once, or floats entirely above or below. Engineers use it to predict feasibility; students use it to check their algebra before it goes wrong.

What complex roots mean physically

A negative discriminant does not make the problem unsolvable; it means no real number satisfies it. In oscillating systems those complex parts encode damping and frequency, which is why electrical engineers treat complex roots as features rather than failures.

Why a = 0 breaks everything

The quadratic formula divides by 2a, so zero there is undefined. Worse, without an x² term the graph is a line with exactly one root −c/b; a different tool for a different shape, and the calculator says so instead of erroring cryptically.

How it's calculated

The math behind this calculator

x = (−b ± √(b² − 4ac)) / 2a Discriminant D = b² − 4ac

The discriminant b² − 4ac is computed first because it classifies everything: positive yields two distinct real roots via the quadratic formula, zero collapses to one repeated root at −b/2a, and negative produces a complex conjugate pair written a ± bi. The axis of symmetry row shows where the parabola pivots.

Assumptions & limitations

  • Coefficient a must be non-zero or the equation is not quadratic.
  • Complex results are formatted as x = real ± imaginary·i.
  • Roots display up to six decimal places.

Worked example

Solving x² − 5x + 6 = 0 gives discriminant 1 and two real roots: x = 3 and x = 2; it factors as (x−3)(x−2).

FAQ

Frequently asked questions

How do I enter a missing term?
Enter 0 for that coefficient; e.g. x² − 9 = 0 uses a=1, b=0, c=−9.
What does “double root” mean?
The parabola touches the x-axis at exactly one point, so both formula branches produce the same value.
Can I get decimal roots?
Yes; irrational roots render to six decimals, enough for checking homework or engineering estimates.
Why is a=0 rejected?
Without the squared term the equation is linear, and the quadratic formula would divide by zero.

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